[Paper Review] Elliptic PDEs with distributional drift and backward SDEs driven by a c{\`a}dl{\`a}g martingale with random terminal time

Francesco Russo, Lukas Wurzer|arXiv (Cornell University)|Jul 11, 2014

Stochastic processes and financial applications16 references4 citations

TL;DR

This paper introduces a generalized class of semilinear elliptic PDEs with distributional drift, establishing existence and uniqueness of $C^1$ generalized solutions. It links these PDEs to backward SDEs driven by a c{\'a}dl\{a}g martingale with random terminal time, proving uniqueness under general conditions on the driving martingale.

ABSTRACT

We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized solutions of class $C^1$. The generator $L$ is associated with a Markov process $X$ which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward process is $X$. Since $X$ is a weak solution of the forward SDE, the BSDE appears naturally to be driven by a martingale. In the paper we also discuss the uniqueness of a BSDE with random terminal time when the driving process is a general c{a}dl{a}g martingale.

Motivation & Objective

• To extend semilinear elliptic PDEs to include distributional drift, broadening the class of admissible generators.
• To establish existence and uniqueness of $C^1$ generalized solutions for such PDEs.
• To connect the PDE solution to a backward SDE with random terminal time driven by a c{\'a}dl{\'a}g martingale.
• To investigate uniqueness of backward SDEs when the driving process is a general c{\'a}dl{\'a}g martingale with random terminal time.

Proposed method

• Formalize a generalized notion of the second-order operator $L$ with distributional drift via weak solutions of SDEs.
• Construct the forward Markov process $X$ as a weak solution to an SDE with distributional drift.
• Define the associated backward SDE with random terminal time, where the forward process is $X$.
• Use the martingale representation property to derive the BSDE driven by a c{\'a}dl{\'a}g martingale.
• Apply functional analytic techniques to prove existence and uniqueness of $C^1$ solutions to the PDE.
• Establish uniqueness of the BSDE solution under general conditions on the c{\'a}dl{\'a}g martingale driver.

Experimental results

Research questions

• RQ1 Can semilinear elliptic PDEs with distributional drift admit $C^1$ generalized solutions, and if so, under what conditions?
• RQ2 How is the solution of such a PDE naturally linked to a backward SDE with random terminal time?
• RQ3 What conditions ensure uniqueness of the solution to a backward SDE driven by a c{\'a}dl{\'a}g martingale with random terminal time?
• RQ4 How does the weak solution structure of the forward SDE with distributional drift affect the BSDE formulation?

Key findings

• The paper establishes existence and uniqueness of $C^1$ generalized solutions for semilinear elliptic PDEs with distributional drift.
• The solution of the PDE is shown to be equivalent to the solution of a backward SDE with random terminal time.
• The BSDE is driven by a c{\'a}dl{\'a}g martingale due to the weak solution nature of the forward SDE.
• Uniqueness of the BSDE solution is proven under general conditions on the c{\'a}dl{\'a}g martingale driver.